Browsing by Subject "Elliptic"
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Item About the largest subsolution for a free boundary problem in R²(2010-12) Orcan, Betul; Caffarelli, Luis A.; Dawson, Clinton; Gamba, Irene; Souganidis, Panagiotis; Ying, LexingWe analyze the geometry and regularity of the largest subsolution of a Free Boundary Problem. We showed that the largest subsolution is a viscosity solution of (1) with Lipschitz and Non-Degenerate properties under a very general free boundary condition. In addition to this, we provide density bounds for the positivity set and its complement near the free boundary.Item Fast direct algorithms for elliptic equations via hierarchical matrix compression(2010-08) Schmitz, Phillip Gordon; Ying, Lexing; Gamba, Irene M.; Ghattas, Omar; Gonzalez, Oscar; Ren, KuiWe present a fast direct algorithm for the solution of linear systems arising from elliptic equations. We extend the work of Xia et al. (2009) on combining the multifrontal method with hierarchical matrices. We offer a more geometric interpretation of that approach, extend it in two dimensions to the unstructured mesh case, and detail an adaptive decomposition procedure for selectively refined meshes. Linear time complexity is shown for a quasi-uniform grid and demonstrated via numerical results for the adaptive algorithm. We also provide an extension to three dimensions with proven linear complexity but a more practical variant with slightly worse scaling is also described.Item Groups defined on elliptic curves without flexes(Texas Tech University, 2004-12) Kelter, Charity CNot availableItem A local-nonlocal transmission problem(2015-05) Kriventsov, Dennis; Caffarelli, Luis A.; Arapostathis, Aristotle; Figalli, Alessio; Vasseur, Alexis F; Zitkovic, GordanWe consider solutions to some elliptic equations which change abruptly across a smooth interface. The main equation of interest, motivated by applications to atmospheric dynamics, is local on one side of this interface and nonlocal on the other, and features a critical nonlinear drift term. The major difficulty of the problem stems from a lack of scale invariance caused by the different orders of the different principal terms. While the existence of weak solutions follows from standard methods, the continuity of them across the interface requires a careful investigation of the scale dependence. The main results are a De Giorgi-Nash-Moser type continuity theorem, an in-depth analysis of the nonlocal analogue of the “transmission condition” satisfied by the frozen-coefficient equation, and a perturbative result for sufficiently smooth interfaces.